Regularity of minimizers for double phase functionals of borderline case with variable exponents

Theorem

(Baroni et al. [7, Theorem 2.4, 6.1]) For n ≥ 2 , N ≥ 1 , p > 1 , and a bounded domain Ω ⊂ R n let u ∈ W 1 , 1 ( Ω ; R N ) be a local minimizer of P log defined in (1.1) and assume that the function a ( ⋅ ) is non-negative and bounded. Let ω ( ⋅ ) be its modulus of continuity in the sense that

and denote

Then

• if N = 1 and l < ∞ , then u ∈ C loc 0 , β ( Ω ; R N ) , for some β ∈ ( 0 , 1 ) ,

• if l = 0 , then u ∈ C loc 0 , β ( Ω ; R N ) , for every β ∈ ( 0 , 1 ) ,

• if ω ( r ) ≤ C r α , for some α ∈ ( 0 , 1 ) and C > 0 , then u ∈ C loc 1 , γ ( Ω ; R N ) , for some γ ∈ ( 0 , 1 ) .

Definition 1.1

A function u ∈ W 1 , 1 ( Ω ) is called to be a local minimizer of ℱ if F ( ⋅ , D u ) ∈ L 1 ( Ω ) and satisfies

for any φ ∈ W loc 1 , 1 ( Ω ) with compact support in Ω .

Theorem 1.2

For a bounded domain Ω ⊂ R n let a ( ⋅ ) and p ( ⋅ ) be functions defined on Ω satisfying (1.4) and (1.5) with l a = l s = 0 . Assume also that a ( ⋅ ) ≥ 0 and that 1 < p 0 , p 3 < ∞ ( p 0 and p 3 are defined in (1.6)).

Let u ∈ W 1 , 1 ( Ω ) be a local minimizer of the functional

Then, we have u ∈ C loc 0 , α ( Ω ) for any α ∈ ( 0 , 1 ) .

Moreover, when both a ( ⋅ ) and p ( ⋅ ) are Hölder continuous, then u ∈ C loc 1 , γ ( Ω ) for some γ ∈ ( 0 , 1 ) .

Remark 1.3

Concerning the statement of the theorem by Baroni-Colombo-Mingione mentioned before and Theorem 1.2, the Hölder continuity of a minimizer u for some exponent can still be proven in the vectorial case N > 1, by requiring that l_a and l_p in (1.5) are sufficiently small. See [4, Remark 3.3], which can be adapted to this situation. See also [5].

Lemma 2.1

Assume that a , b , c , α , β , γ satisfy (2.6) and that f ∈ L a log α L ( B R ( y ) ) , where R ∈ ( 0 , 1 ) . Then, there exists a constant C = C ( α , β , γ , a , b , c , n ) for which the following estimate holds:

Proof

By (2.2), (2.4), and (2.5) with g = 1 , we have that

where we write simply ‖ ⋅ ‖ L p ( p = a , b , c ) for ‖ ⋅ ‖ L p ( B R ( y ) ) .

By (2.2), we can estimate the first integral of the right-hand side of (2.8) as

The second integral can be estimated as follows:

where we used (2.2) and the following fundamental fact:

Combining (2.8)–(2.9), we obtain

Using “standard" Hölder inequality, we have that

Now, using this inequality and mentioning that

holds for some constant M = M ( α , β , a , b , c , n ) , we obtain (2.7) from (2.10).□

Lemma 3.1

Let A , B , α be positive constants and β ∈ ( 0 , α ) . Then, for any γ < α , there exists a positive constant ε 0 = ε 0 ( α , β , γ , A ) with the following property: let us consider a non-negative and nondecreasing function Φ defined on [ 0 , R 0 ] for some R 0 > 0 satisfying, for ε ∈ ( 0 , ε 0 ) , the following inequality:

for all 0 < r < R 0 . Then, for some positive constant c ( α , β , γ , A )

holds for any r ∈ ( 0 , R 0 ] .

Lemma 3.2

Let Φ : [ r 1 , r 2 ] → [ 0 , ∞ ) be a bounded function. Assume that for r 1 ≤ s < t ≤ r 2 we have

with A , B ≥ 0 and 0 ≤ θ < 1 . Then, there exists a positive constant c ( γ , θ ) depending only on γ and θ such that

Proposition 3.3

Let a ( ⋅ ) and ω a ( ⋅ ) be as in (1.4) with (1.5), and H γ ( ⋅ , ⋅ ) be as (3.2) with p > 1 and γ ∈ [ 1 , p ) . Put ω 0 ≔ ω a ( e − p ∕ n ) . Then, there exists a constant C 0 > 0 depending only on  data  and l of (1.5), such that

holds for any f ∈ L p ( B R ( x ) ) , where B R ( x ) ⊂ R n is a ball with R ≤ e − p ∕ n .

Proof

We can prove this proposition keeping in mind [12, Proposition 3.1]. Put

Since H γ , R is convex, Jensen’s inequality gives

Now, we observe that

Since, by Hölder’s inequality we have

for some positive constant C ( n ) depending only on n , taking R sufficiently small so that R − n ∕ p ≥ e and using (2.1), we obtain

Combining the above estimate with (3.5), we obtain, for some constant C ( γ ) depending only on γ , that

Now, mentioning condition (1.5) and combining the above estimate with (3.4), we obtain (3.3).□

Remark 3.4

In [12], Proposition 3.1 is corresponding to the above one for H ( x , z ) = ∣ z ∣ p + a ( x ) ∣ z ∣ q and is applied to H ¯ ( x , z ) = ∣ z ∣ p ∕ γ + [ a ( x ) ] 1 ∕ γ ∣ z ∣ q ∕ γ . This is possible since the Hölder continuity of a 1 ∕ γ ( ⋅ ) is guaranteed by that of a ( ⋅ ) . However, in our case we are supposing only logarithmic continuity on a ( ⋅ ) pointing out that it does not imply the logarithmic continuity of a 1 ∕ γ ( ⋅ ) . By such a reason, in this article, we show the above proposition for H γ directly.

Theorem 3.5

Let a ( ⋅ ) and ω a be as in (1.4) with condition (1.5), and H ( ⋅ , ⋅ ) be as in (3.1) with p > 1 . Let μ ∈ L ∞ ( R n ) , μ ≥ 0 , and such that ∫ B R μ ( x ) d x = 1 , for B R ⊂ Ω with R ∈ ( 0 , 1 ) . For μ and u ∈ L 1 ( B R ) , let us denote

Then, there exist exponents d 1 , d 2 : d 1 > 1 > d 2 depending only on n , p , such that

holds, for every B R ≔ B R ( y ) ⊂ Ω with R < r 0 ( p ) and u ∈ W 1 , 1 ( Ω , R N ) with [ H ( x , D u ) ] d 2 ∈ L loc 1 ( Ω ) . The constant C depends only on  data, sup μ , and ‖ D u ‖ L p ( B R ( x 0 ) ) .

Proof

Using (3.3), instead of [12, (3.1)], we can proceed as in the proof of [12, Theorem 1.6].

Step 1: Maximal estimate.

For bounded domain Ω ⊂ R n and f ∈ L 1 ( Ω , R k ) , we define the restricted maximal operator:

For 1 < p ≤ ∞ , 1 ≤ γ < p and f ∈ L p ∕ γ ( Ω ; R k ) , we have the following Hardy-Littlewood maximal inequality (of strong type):

We show the following estimate corresponding to [12, (3.4)], but for H γ , by the motivation stated in Remark 3.4. For γ ∈ ( 1 , p ) , let H γ be the function defined in (3.2). Then, we show the following estimate:

Let us set γ ′ ∈ ( γ , p ) . Applying (3.3) for H γ ′ on every B r ( x ) , taking supremum and mentioning the monotonicity of the function t ↦ H γ ′ ( ⋅ , t ) , we obtain